Engineering Mathematics

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Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Received: Jun. 01, 2019    Accepted: Jul. 08, 2019    Published: Jul. 17, 2019
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Abstract

This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler–Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.

DOI 10.11648/j.engmath.20190301.15
Published in Engineering Mathematics ( Volume 3, Issue 1, June 2019 )
Page(s) 19-29
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Cracked Nanobeam, Free Vibration, Euler–Bernoulli Theory, Timoshenko Theory, Differential Quadrature Method

References
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[2] Eringen, A. C., Nonlocal polar elastic continua. International journal of engineering science, 1972. 10(1): p. 1-16.
[3] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of applied physics, 1983. 54(9): p. 4703-4710.
[4] Eringen, A. C., Nonlocal continuum field theories. 2002: Springer Science & Business Media.
[5] Eringen, A. C. and D. G. B. Edelen, On nonlocal elasticity. International Journal of Engineering Science, 1972. 10(3): p. 233-248.
[6] Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures, 2009. 41(9): p. 1651-1655.
[7] Reddy, J., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 2007. 45(2-8): p. 288-307.
[8] Wang, C. M., Y. Y. Zhang, and X. Q. He, Vibration of nonlocal Timoshenko beams. Nanotechnology, 2007. 18(10): p. 105401.
[9] Behera, L. and S. Chakraverty, Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials. Applied Nanoscience, 2014. 4(3): p. 347-358.
[10] Wu, L.-Y., et al., Vibrations of nonlocal Timoshenko beams using orthogonal collocation method. Procedia Engineering, 2011. 14: p. 2394-2402.
[11] Eltaher, M., A. E. Alshorbagy, and F. Mahmoud, Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 2013. 37(7): p. 4787-4797.
[12] Beni, Y. T., A. Jafaria, and H. Razavi, Size effect on free transverse vibration of cracked nano-beams using couple stress theory. International Journal of Engineering-Transactions B: Applications, 2014. 28(2): p. 296-304.
[13] Hasheminejad, S. M., et al., Free transverse vibrations of cracked nanobeams with surface effects. Thin Solid Films, 2011. 519(8): p. 2477-2482.
[14] Loghmani, M. and M. R. Hairi Yazdi, An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach. Results in Physics, 2018. 11: p. 166-181.
[15] Roostai, H. and M. Haghpanahi, Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Applied Mathematical Modelling, 2014. 38(3): p. 1159-1169.
[16] Loya, J., et al., Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model. Journal of Applied Physics, 2009. 105(4): p. 044309.
[17] Sourki, R. and S. Hoseini, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory. Applied Physics A, 2016. 122(4): p. 413.
[18] Sourki, R. and S. Hosseini, Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam. The European Physical Journal Plus, 2017. 132(4): p. 184.
[19] Bahrami, A., A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams. Composites Part B: Engineering, 2017. 120: p. 168-181.
[20] Wang, K. and B. Wang, Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. Journal of Vibration and Control, 2015. 21(12): p. 2452-2464.
[21] Torabi, K. and J. Nafar Dastgerdi, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model. Thin Solid Films, 2012. 520(21): p. 6595-6602.
[22] Soltanpour, M., et al., Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium. Microsystem Technologies, 2017. 23(6): p. 1813-1830.
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[24] Osman, T., et al., Applied and Computational Mathematics Analysis of cracked plates using localized multi-domain differential quadrature method. Vol. 2. 2013. 109-114.
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  • APA Style

    Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. (2019). Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Engineering Mathematics, 3(1), 19-29. https://doi.org/10.11648/j.engmath.20190301.15

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    ACS Style

    Mohamed Abd Elkhalek; Tharwat Osman; Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng. Math. 2019, 3(1), 19-29. doi: 10.11648/j.engmath.20190301.15

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    AMA Style

    Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng Math. 2019;3(1):19-29. doi: 10.11648/j.engmath.20190301.15

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  • @article{10.11648/j.engmath.20190301.15,
      author = {Mohamed Abd Elkhalek and Tharwat Osman and Mohamed Saad Matbuly},
      title = {Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique},
      journal = {Engineering Mathematics},
      volume = {3},
      number = {1},
      pages = {19-29},
      doi = {10.11648/j.engmath.20190301.15},
      url = {https://doi.org/10.11648/j.engmath.20190301.15},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.engmath.20190301.15},
      abstract = {This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler–Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.},
     year = {2019}
    }
    

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  • TY  - JOUR
    T1  - Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique
    AU  - Mohamed Abd Elkhalek
    AU  - Tharwat Osman
    AU  - Mohamed Saad Matbuly
    Y1  - 2019/07/17
    PY  - 2019
    N1  - https://doi.org/10.11648/j.engmath.20190301.15
    DO  - 10.11648/j.engmath.20190301.15
    T2  - Engineering Mathematics
    JF  - Engineering Mathematics
    JO  - Engineering Mathematics
    SP  - 19
    EP  - 29
    PB  - Science Publishing Group
    SN  - 2640-088X
    UR  - https://doi.org/10.11648/j.engmath.20190301.15
    AB  - This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler–Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.
    VL  - 3
    IS  - 1
    ER  - 

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Author Information
  • Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

  • Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

  • Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt

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